Double-Peak Splitting: The Hidden Structure of Finite-Lattice Phase Transitions in the 3D Ising Model——Verification of the Factor Hierarchy Law on an NP-Complete System: Double-Peak Splitting, Transition Regime, Piecewise Control Function, and Multi-Regime State
On the L=64 lattice of the 3D Ising model—an NP-complete problem rigorously proven to have no analytical solution—the six functional dimensions of the Testability Norms were applied, achieving the detection and classification of the ferromagnetic-paramagnetic phase transition with a statistical confidence of p = 3.33×10⁻⁷⁰. Core findings: The Bai-Perron double-break blind test automatically locked T₁ = 4.39 and T₂ = 4.49 (F = 285.7, significantly outperforming the single-break F = 238.3), precisely aligned with the M_sp switch (4.40, F = 1524) and the U₄ switch (4.50, F = 2335), with deviations of 0.01 and 0.01 respectively. This reveals the Double-Peak Splitting of second-order phase transitions on finite lattices—between two sequential regime switches lies an independent Transition Regime (width approximately 0.1), a region commonly dismissed in standard finite-size scaling (FSS) analysis practice as "rounded crossover requiring extrapolation-based elimination," whose boundaries are algorithmically identifiable. The structure was cross-size replicated on the 2D Potts q=4 model: double-break models significantly outperformed single-break models at L = 16, 32, and 64, with the peak separation shrinking from 0.260 at L = 64 to 0.180 at L = 128, evolving according to scaling laws. The traditional gold standard for phase transition detection, magnetic susceptibility χ, fell completely silent under the Chow test (F = 4.8), yet the 1/χ transformation activated it to 324—revealing that the Chow test requires alignment with the correct mathematical structure to capture physical signals. The regime switch temperature in the M_sp dimension remained stable across all lattice sizes (L = 16, 32, 64, 128) at 4.40–4.46, showing no convergence toward the thermodynamic limit Tc = 4.5115 as L increases—pointing to a Precursor Regime Boundary independent of the thermodynamic limit, cross-model replicated in the M_sp dimension of Potts q=4. The first-order transition (Potts q=3) was confirmed as the direction-inverting type (intercept-shift dominated), and the KT transition (XY model) as single-stage continuous crossover—together with the Double-Peak Splitting of second-order transitions, forming a three-category classification framework of the Factor Hierarchy Law. A three-segment piecewise control function constructed from the Bai-Perron double breaks provides, for this NP-complete system, a quantitative description refined directly from data that is functionally equivalent to the Onsager exact solution for the 2D Ising model—requiring no free energy function. The two inflection points of this function were automatically identified by the Bai-Perron algorithm, without relying on any a priori definition of free energy or order parameter, directly demonstrating the capacity of the Testability Norms to extract physical laws from data. Methodological contribution and ultimate asset: This paper distills the entire body of verification into the Multi-Regime State Decoupling Algorithm (MRSD) based on the Factor Hierarchy Law. Its core mathematical logic proceeds in three steps: (1) input any complex high-dimensional dataset; (2) execute the structural optimization operator \left\{T_k\right\}=arg\min_{}{\sum\nolimits^{k+1}_{j=1}{}}\mathrm{RSS}_j(T)+λ⋅k; (3) output a dynamic regime allocation matrix, informing the user at which point the system logic undergoes "rule resetting" and at which point the structure enters "rigid locking." The algorithm has been cross-model verified on the 3D Ising, 2D Potts q=4, and 2D XY models, requiring no free energy function or a priori order parameter definition—only that the data be objective. It redefines finite-size effects from "errors requiring correction" to independent statistical regimes with rigorous boundaries that are algorithmically identifiable, completing the key leap of the information-theoretic duality between the Factor Hierarchy Law and the Ehrenfest classification from analytically solvable systems to systems without analytical solutions. The Testability Norms achieved full verification closure on an NP-complete system across all six functional dimensions. Leave-one-out cross-validation (Tc range [4.47, 4.48], standard deviation 0.0021), Bootstrap threshold regression (95% CI [4.44, 4.48]), and the shrinking external field convergence verification (Tc monotonically shifted from 4.48 to 4.50) jointly confirm the high robustness of the conclusions. Research Paradigm Statement: The core methodology, research direction, and final decisions were independently directed by the author. DeepSeek assisted with code implementation, data presentation, and text drafting. The author takes full academic responsibility for the final content.
On the L=64 lattice of the 3D Ising model—an NP-complete problem rigorously proven to have no analytical solution—the six functional dimensions of the Testability Norms were applied, achieving the detection and classification of the ferromagnetic-paramagnetic phase transition with a statistical confidence of p = 3.33×10⁻⁷⁰. Core findings: The Bai-Perron double-break blind test automatically locked T₁ = 4.39 and T₂ = 4.49 (F = 285.7, significantly outperforming the single-break F = 238.3), precisely aligned with the M_sp switch (4.40, F = 1524) and the U₄ switch (4.50, F = 2335), with deviations of 0.01 and 0.01 respectively. This reveals the Double-Peak Splitting of second-order phase transitions on finite lattices—between two sequential regime switches lies an independent Transition Regime (width approximately 0.1), a region commonly dismissed in standard finite-size scaling (FSS) analysis practice as "rounded crossover requiring extrapolation-based elimination," whose boundaries are algorithmically identifiable. The structure was cross-size replicated on the 2D Potts q=4 model: double-break models significantly outperformed single-break models at L = 16, 32, and 64, with the peak separation shrinking from 0.260 at L = 64 to 0.180 at L = 128, evolving according to scaling laws. The traditional gold standard for phase transition detection, magnetic susceptibility χ, fell completely silent under the Chow test (F = 4.8), yet the 1/χ transformation activated it to 324—revealing that the Chow test requires alignment with the correct mathematical structure to capture physical signals. The regime switch temperature in the M_sp dimension remained stable across all lattice sizes (L = 16, 32, 64, 128) at 4.40–4.46, showing no convergence toward the thermodynamic limit Tc = 4.5115 as L increases—pointing to a Precursor Regime Boundary independent of the thermodynamic limit, cross-model replicated in the M_sp dimension of Potts q=4. The first-order transition (Potts q=3) was confirmed as the direction-inverting type (intercept-shift dominated), and the KT transition (XY model) as single-stage continuous crossover—together with the Double-Peak Splitting of second-order transitions, forming a three-category classification framework of the Factor Hierarchy Law. A three-segment piecewise control function constructed from the Bai-Perron double breaks provides, for this NP-complete system, a quantitative description refined directly from data that is functionally equivalent to the Onsager exact solution for the 2D Ising model—requiring no free energy function. The two inflection points of this function were automatically identified by the Bai-Perron algorithm, without relying on any a priori definition of free energy or order parameter, directly demonstrating the capacity of the Testability Norms to extract physical laws from data. Methodological contribution and ultimate asset: This paper distills the entire body of verification into the Multi-Regime State Decoupling Algorithm (MRSD) based on the Factor Hierarchy Law. Its core mathematical logic proceeds in three steps: (1) input any complex high-dimensional dataset; (2) execute the structural optimization operator \left{T_k\right}=arg\min_{}{\sum\nolimits^{k+1}_{j=1}{}}\mathrm{RSS}_j(T)+λ⋅k; (3) output a dynamic regime allocation matrix, informing the user at which point the system logic undergoes "rule resetting" and at which point the structure enters "rigid locking." The algorithm has been cross-model verified on the 3D Ising, 2D Potts q=4, and 2D XY models, requiring no free energy function or a priori order parameter definition—only that the data be objective. It redefines finite-size effects from "errors requiring correction" to independent statistical regimes with rigorous boundaries that are algorithmically identifiable, completing the key leap of the information-theoretic duality between the Factor Hierarchy Law and the Ehrenfest classification from analytically solvable systems to systems without analytical solutions. The Testability Norms achieved full verification closure on an NP-complete system across all six functional dimensions. Leave-one-out cross-validation (Tc range [4.47, 4.48], standard deviation 0.0021), Bootstrap threshold regression (95% CI [4.44, 4.48]), and the shrinking external field convergence verification (Tc monotonically shifted from 4.48 to 4.50) jointly confirm the high robustness of the conclusions. Research Paradigm Statement: The core methodology, research direction, and final decisions were independently directed by the author. DeepSeek assisted with code implementation, data presentation, and text drafting. The author takes full academic responsibility for the final content.
Replication Data and Code for: "Capturing the Universality Class of the 2D Ising Model via Symmetry-Corrected Machine Learning and Finite-Size Scaling" This repository contains the complete source code, simulation datasets, and analysis scripts required to reproduce all findings presented in the associated manuscript. The study introduces a symmetry-corrected machine learning pipeline (Gauge-Fixed PCA + Logistic Regression) to identify phase transitions and estimate critical temperatures with high precision, validated on the two-dimensional Ising model. Repository Contents: 01_generate_data.py — Generates 2D Ising model spin configurations using the Wolff cluster algorithm. Produces equilibrium datasets for lattice sizes L∈{32,48,64,96}L \in \{32, 48, 64, 96\} L∈{32,48,64,96} with 10,000 independent configurations per temperature point. 02_analysis_pipeline_v2.py — Core analysis script: performs gauge fixing (σ′=σ⋅sgn(m)\sigma' = \sigma \cdot \mathrm{sgn}(m) σ′=σ⋅sgn(m)), PCA dimensionality reduction (d=16d = 16 d=16, cross-validated), confusion-scheme classification with parabolic sub-grid interpolation, bootstrap uncertainty quantification (Nboot=50N_{\rm boot} = 50 Nboot=50), and pFMp_{\rm FM} pFM computation with isotonic calibration. Outputs summary_ml_v2.json. 03_plot_main_figures.py — Reproduces Figures 1–6 of the main manuscript. 04_plot_supplementary_v2.py — Reproduces Supplementary Figures S1–S5 and Table 1. ablation_preprocessing_S5.py — Standalone preprocessing ablation script: runs raw and ∣m∣|m| ∣m∣-only comparisons and generates Supplementary Figure S5. Does not modify main results. summary_ml_v2.json — Pre-computed results (gauge-fixed pipeline) for immediate reproduction of all main figures without re-running Monte Carlo simulations. summary_ml_v2_raw.json — Pre-computed ablation results: raw (no gauge fixing) preprocessing, L∈{32,64,96}L \in \{32, 64, 96\} L∈{32,64,96}. summary_ml_v2_absm.json — Pre-computed ablation results: ∣m∣|m| ∣m∣-only preprocessing, L∈{32,64,96}L \in \{32, 64, 96\} L∈{32,64,96}. Table1_Tc_Comparison.tex — LaTeX source for Table 1. requirements.txt — Python dependencies. README.txt — Step-by-step reproduction instructions. Methodology Highlights: System sizes: L∈{32,48,64,96}L \in \{32, 48, 64, 96\} L∈{32,48,64,96} 10,000 independent configurations per temperature point; Wolff cluster algorithm with 10410^4 104 thermalization steps Symmetry correction: global Z2\mathbb{Z}_2 Z2 gauge fixing prior to PCA Confusion scheme: validation cross-entropy minimization over T0∈[2.10,2.45]T_0 \in [2.10, 2.45] T0∈[2.10,2.45] (60 candidate values, band Δ=0.05\Delta = 0.05 Δ=0.05, 5-fold CV) Bootstrap: 50 configuration-level resamples; parabolic sub-grid interpolation for discretization bias reduction Key result: Tc≈2.2717(8)T_c \approx 2.2717(8) Tc≈2.2717(8) for L=96L = 96 L=96 (0.6%0.6\% 0.6% from Onsager); universality class confirmed via FSS data collapse with ν=1\nu = 1 ν=1
We study the Ising phase transition on a three-dimensional cubic lattice augmented with sparse, distance-dependent shortcuts whose density decays as P(r) proportional to r^(-2). This network ensemble, characterized by geometric embedding, sparse additive quenched shortcuts, and power-law distance dependence, falls outside the scope of existing results for both uniform small-world networks (Barrat-Weigt, Herrero) and dense long-range Ising models (Fisher-Ma-Nickel, Sak). We present partial analytical results. The effective propagator exhibits three regimes: a Wilson-Fisher intermediate regime (standard Laplacian dominant), a Levy IR regime (wormhole term dominant), and a biharmonic UV regime, with propagator crossover at approximately 125 lattice units. A scaling estimate suggests the nonlocal operator is only marginally relevant at Wilson-Fisher, implying a potentially very slow RG flow. A mean free path analysis yields a finite effective signal velocity of approximately 2.4 times the local propagation speed, with the exponent alpha = 2 lying at the apparent boundary for finite-speed propagation on sparse shortcut networks. We propose a concrete Monte Carlo simulation protocol (approximately 500 CPU-hours) using Wolff cluster algorithm and finite-size scaling with Binder cumulant analysis to definitively determine the universality class. The key observable is U4*, which unambiguously distinguishes 3D Ising (U4* approximately 0.61) from mean-field (U4* approximately 0.47) universality. Control simulations on pure 3D lattice and uniform shortcut networks are specified. The problem is relevant to discrete quantum gravity models with holographic connectivity, network science, and the theory of phase transitions on complex geometries. All analytical results are clearly labeled as scaling estimates or conjectures where appropriate, and five explicit unresolved issues are identified.
Two temperature scales in the Ising model on the {5,4} hyperbolic lattice with free boundaries: susceptibility peak and boundary-induced order
On the {5,4} hyperbolic lattice, the outermost generation holds 73\% of the sites at all system sizes, making macroscopic averages strictly boundary-dependent. We study the ferromagnetic Ising model on this geometry with free boundaries by Monte Carlo simulation, demonstrating that conventional observables cease to identify a single critical scale. Instead, the finite lattices are organized by two distinct temperature scales. The susceptibility maximum identifies the lower transition scale at Tc2=1.4782(15)J/kB without extrapolation. However, the order-parameter distribution lacks a conventional fixed point at this scale. A distinct pseudocritical scale emerges instead at T∗≃1.67(3) inside the boundary-sensitive intermediate phase, where Binder cumulant curves cross and effective exponents are compatible with mean-field criticality. Between Tc2 and T∗, the interior amplifies boundary fluctuations into induced order. Furthermore, because the volume grows exponentially with depth, finite-size scaling must be formulated in the generation index rather than the total number of sites. The lack of asymptotic power-law convergence in the scaling stretch confirms through an independent observable that T∗ is a finite-size crossover rather than a thermodynamic fixed point. Finally, by applying a coherent field to the outermost generation alone, we recover the upper thermodynamic transition at Tpt=2.81(1)J/kB, demonstrating that it remains accessible under an appropriate boundary perturbation.
We employ the microcanonical inflection-point analysis method, developed for the systematic identification and classification of phase transitions in systems of any size, to study the two-dimensional Ising model at various lattice sizes and in the thermodynamic limit. Exact results for the density of states, which were obtained by exact algorithmic computation, provide evidence for higher-order transitions in addition to the well-studied second-order ferromagnetic-paramagnetic phase transition. An independent third-order phase transition is identified in the ferromagnetic phase, whereas another third-order transition resides in the paramagnetic phase. The latter is a dependent transition, i.e., it is inevitably associated with the critical transition, but it remains separate from the critical point in the thermodynamic limit. For a deeper insight into the nature of these additional transitions, a detailed analysis of spin clusters is performed.
To demonstrate the application of computational physics in modern statistical physics, this paper investigates the critical behavior of a finite-size ferromagnetic system based on the two-dimensional square-lattice Ising model.A nearest-neighbor Ising Hamiltonian is constructed, and periodic boundary conditions are adopted to reduce finite-boundary effects.The Metropolis Monte Carlo algorithm is used to sample spin configurations at different temperatures, while energy, magnetization, specific heat, magnetic susceptibility and Binder cumulant are calculated to describe the transition from an ordered state to a disordered state.The simulation results show that the system exhibits clear long-range order at low temperatures, where the order parameter remains high.As temperature increases, thermal fluctuations become stronger, magnetization decreases rapidly, and the system gradually evolves into a disordered state.The specific heat curve shows a pronounced peak at approximately (T = 2.269), while the susceptibility peak shifts slightly toward the high-temperature side due to finite-size effects and the discrete temperature grid.Overall, the simulated critical region is consistent with the theoretical critical temperature of the two-dimensional square-lattice Ising model.Spin-configuration analysis further reveals enhanced fluctuations and domain reconstruction near the critical region.In addition, a thermalization criterion and an extended sampling test for L = 48 near the critical region are added to examine the influence of finite sampling and equilibration.These results indicate that Monte Carlo simulation of the two-dimensional Ising model can effectively reproduce continuous phase transition, critical fluctuation and finite-size effects, providing a basic computational case for understanding critical phenomena in complex systems and data-driven physical analysis.